Problem: All of the 5th grade teachers and students from Oak went on a field trip to an archaeology museum. Tickets were $$7.00$ each for teachers and $$4.50$ each for students, and the group paid $$52.50$ in total. The next month, the same group visited a science museum where the tickets cost $$14.00$ each for teachers and $$9.50$ each for students, and the group paid $$108.50$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7x+4.5y = 52.5}$ ${14x+9.5y = 108.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-14x-9y = -105}$ ${14x+9.5y = 108.5}$ Add the top and bottom equations together. $ 0.5y = 3.5 $ $ y = \dfrac{3.5}{0.5}$ ${y = 7}$ Now that you know ${y = 7}$ , plug it back into $ {7x+4.5y = 52.5}$ to find $x$ ${7x + 4.5}{(7)}{= 52.5}$ $7x+31.5 = 52.5$ $7x = 21$ $x = \dfrac{21}{7}$ ${x = 3}$ You can also plug ${y = 7}$ into $ {14x+9.5y = 108.5}$ and get the same answer for $x$ ${14x + 9.5}{(7)}{= 108.5}$ ${x = 3}$ There were $3$ teachers and $7$ students on the field trips.